Embryonic development leads to the reproducible and ordered appearance of complexity from egg to adult. The successive differentiation of different cell types that elaborate this complexity results from the activity of gene networks and was likened by Waddington to a flow through a landscape in which valleys represent alternative fates. Geometric methods allow the formal representation of such landscapes and codify the types of behaviors that result from systems of differential equations. Results from Smale and coworkers imply that systems encompassing gene network models can be represented as potential gradients with a Riemann metric, justifying the Waddington metaphor. Here, we extend this representation to include parameter dependence and enumerate all three-way cellular decisions realizable by tuning at most two parameters, which can be generalized to include spatial coordinates in a tissue. All diagrams of cell states vs. model parameters are thereby enumerated. We unify a number of standard models for spatial pattern formation by expressing them in potential form (i.e., as topographic elevation). Turing systems appear nonpotential, yet in suitable variables the dynamics are low dimensional and potential. A time-independent embedding recovers the original variables. Lateral inhibition is described by a saddle point with many unstable directions. A model for the patterning of the Drosophila eye appears as relaxation in a bistable potential. Geometric reasoning provides intuitive dynamic models for development that are well adapted to fit time-lapse data.

胚胎发育导致从卵到成人的复杂性的可重复和有序外观。阐述这种复杂性的不同细胞类型的连续分化是由基因网络的活动造成的，Waddington 将其比作流经山谷代表不同命运的景观。几何方法允许对此类景观进行形式化表示，并将由微分方程系统产生的行为类型编码。Smale 和同事的结果表明，包含基因网络模型的系统可以表示为具有 Riemann 度量的潜在梯度，从而证明了 Waddington 隐喻的合理性。在这里，我们将此表示扩展为包括参数依赖性并枚举所有通过调整最多两个参数可实现的三向细胞决策，这可以概括为包括组织中的空间坐标。因此列举了所有细胞状态与模型参数的关系图。我们通过以潜在形式（即地形高程）表示它们来统一空间模式形成的许多标准模型。图灵系统似乎是无势的，但在合适的变量中，动力学是低维和潜在的。与时间无关的嵌入可以恢复原始变量。横向抑制由具有许多不稳定方向的鞍点描述。一个模型的图案 图灵系统似乎是无势的，但在合适的变量中，动力学是低维和潜在的。与时间无关的嵌入可以恢复原始变量。横向抑制由具有许多不稳定方向的鞍点描述。一个模型的图案 图灵系统似乎是无势的，但在合适的变量中，动力学是低维和潜在的。与时间无关的嵌入可以恢复原始变量。横向抑制由具有许多不稳定方向的鞍点描述。一个模型的图案果蝇眼睛在双稳态电位中表现为松弛。几何推理为开发提供了直观的动态模型，这些模型很好地适应了延时数据。